On the vanishing of Green's function, desingularization and Carleman's method

Abstract

The subject of the present paper is the phenomenon of vanishing of the Green function of the operator $-\Delta +V$ on ${\mathbb{R}}^3$ at the points where a potential $V$ has positive critical singularities. More precisely, imposing minimal assumptions on $V$ (i.e. the form-boundedness), we obtain an upper bound on the order of vanishing of the Green function. As a by-product of our proof, we improve the existing results on the strong unique continuation for eigenfunctions of $-\Delta +V$ in dimension $d=3$.

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